Finite Group Databasebeta

← #78012

Subgroups — #78012

Order 384.

408 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#398C211yesmaximal, characteristic, complemented
2#4994C212yesmaximal, complemented
3#4991C212yesmaximal
4#4988C211yesmaximal, characteristic
5#4980C212yesmaximal, complemented
6#4979C211yesmaximal, characteristic, complemented
7#4675C212yesmaximal, complemented
8#4674C212yesmaximal, complemented
9#4673C212yesmaximal, complemented
10#2830—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×(C2×D3)⋊C4C2212yes—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×C4⋊Dic3C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C3×#95C2212yes—
26C3×#95C2211yescharacteristic
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2211yescharacteristic
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32#398D311yescharacteristic, complemented
33#397—32no—
34#397—32no—
35#396—32no—
36#395—31no—
37#394—32no—
38#393—32no—
39#390—31no—
40#387—32no—
41C23×D3C2312yes—
42C22×C3×C4C2312yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C2×D12—22no—
51C2×D12—41no—
52C2×D12—41no—
53C2×Dic6—42no—
54C2×Dic6—41no—
55C3×C4⋊C4—24no—
56C3×C4⋊C4—41no—
57C3×C4⋊C4—41no—
58(C2×D3)⋊C4—42no—
59(C2×D3)⋊C4—42no—
60(C2×D3)⋊C4—42no—
61(C2×D3)⋊C4—42no—
62(C2×D3)⋊C4—42no—
63(C2×D3)⋊C4—42no—
64C4⋊Dic3—42no—
65Dic3⋊C4—42no—
66C4×Dic3—41no—
67C22×Q8—31no—
68C22×D4—31no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4C2×D312yes—
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C4⋊C4C2×D311yescharacteristic
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C42—31no—
81#95—32no—
82#95—32no—
83#95—32no—
84#95—32no—
85#95—31no—
86#95C2×D312yes—
87#95C2×D311yescharacteristic
88C23×C3C2411yescharacteristic
89C22×D3—22no—
90C22×D3—22no—
91C22×D3—22no—
92C22×D3—22no—
93C22×D3—22no—
94C22×D3—22no—
95C22×D3—22no—
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—22no—
108C2×C3×C4#4112yes—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118D12—44no—
119Dic6—44no—
120C24—32no—
121C2×Q8—121no—
122C2×Q8—122no—
123C2×D4—62no—
124C2×D4—121no—
125C2×D4—121no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—32no—
129C22×C4C22×D312yes—
130C22×C4C22×D311yescharacteristic
131C22×C4C22×D312yes—
132C22×C4C22×D311yescharacteristic
133C22×C4C22×D311yescharacteristic
134C4⋊C4—24no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—122no—
138C4⋊C4—122no—
139C22⋊C4—122no—
140C22⋊C4—122no—
141C22⋊C4—122no—
142C22⋊C4—122no—
143C22⋊C4—122no—
144C22⋊C4—122no—
145C42—121no—
146C22×C3C2×#4111yescharacteristic
147C22×C3H(5,𝔽2)11yescharacteristic
148C22×C3H(5,𝔽2)11yescharacteristic
149C22×C3H(5,𝔽2)12yes—
150C22×C32−1+412yes—
151C2×D3—42no—
152C2×D3—42no—
153C2×D3—42no—
154C2×D3—42no—
155C2×D3—42no—
156C2×D3—42no—
157C2×D3—42no—
158C3×C4—22no—
159C3×C4—42no—
160C3×C4—42no—
161C3×C4—41no—
162C3×C4—41no—
163Dic3—42no—
164Dic3—42no—
165Dic3—42no—
166C23—62no—
167C23—62no—
168C23—62no—
169C23—62no—
170C23—62no—
171C23—62no—
172C23—62no—
173C23C23×D311yescharacteristic
174Q8—124no—
175D4—124no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4#23712yes—
198C2×C3#53811yescharacteristic
199C2×C3#54212yes—
200C2×C3#56011yescharacteristic
201C2×C3#56212yes—
202C2×C3#56311yescharacteristic
203D3—82no—
204C22—122no—
205C22—122no—
206C22—122no—
207C22—122no—
208C22—122no—
209C22—122no—
210C22—122no—
211C22C2×#23711yescharacteristic
212C22#100411yescharacteristic
213C22#100712yes—
214C22#100912yes—
215C22#100911yescharacteristic
216C4—22no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—42no—
221C4—122no—
222C4—122no—
223C4—122no—
224C3#283011yescharacteristic, complemented, Sylow
225C2—242no—
226C2#561711yescharacteristic
227C2#564312yes—
228C2#566611yescharacteristic
229C2#570012yes—
230C2#573511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radicalC22×D12C22
hypercenter#398D3
hypocenterC3#2830
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#398C2semidirect
2. #4994C2semidirect
5. #4980C2semidirect
6. #4979C2semidirect
7. #4675C2semidirect
8. #4674C2semidirect
9. #4673C2semidirect
32. #398D3semidirect
224. C3#2830semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#3982
#49942
#49912
#49882
#49802
#49792
#46752
#46742
#46732
#28303